End-point maximal regularity for the discrete parabolic Cauchy problem and regularity of non-local operators in discrete Besov spaces

In this paper, we study the end-point maximal regularity for the discrete parabolic Cauchy problem \[ \begin{cases} u’(t) + Au(t) = f(t), & t > 0, \cr u(0) = 0, \end{cases} \] where \(-A\) is the discrete Laplacian on the integers \(\mathbb{Z}\). We consider the problem in the space \(\ell^1(\mathbb{Z})\), representing the end-point case where the classical maximal regularity fails in the continuous setting.

We introduce and characterize the discrete Besov spaces \(B^{\alpha}_{p,q}(\mathbb{Z})\) and prove that the operator \(A\) admits maximal regularity in these spaces. Furthermore, we analyze the regularity properties of the fractional powers \(A^\alpha\) and their associated non-local operators.

Our results provide a complete picture of the mapping properties of the discrete heat semigroup in the scale of Besov spaces, generalizing previous results known for the Euclidean space \(\mathbb{R}^n\).

Recommended citation: Abadias, L., De León-Contreras, M., Mahillo, A. (2025). End-point maximal regularity for the discrete parabolic Cauchy problem and regularity of non-local operators in discrete Besov spaces. J. Differential Equations, 440, Paper No. 113465.
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